Question

(a²-b²)(a²+b²)- (a²-b²)²​

Answer 1

» Question :

Simplify :

$\green{\sf{\big(a^{2} - b^{2}\big)\big(a^{2} + b^{2}\big) - \big(a^{2} + b^{2}\big)^{2}}}$

» We Know :

• $\sf{a^{2} - b^{2} = (a + b)(a - b)}$

• $\sf{a^{2} - b^{2} = a^{2} + b^{2} - 2ab}$

» Solution :

$\purple{\sf{\big(a^{2} - b^{2}\big)\big(a^{2} + b^{2}\big) - \big(a^{2} + b^{2}\big)^{2}}}$

Using the identity ,

$\sf{a^{2} - b^{2} = (a + b)(a - b)}$

$\sf{a^{2} - b^{2} = a^{2} + b^{2} - 2ab}$

We Get :

$\sf{\Rightarrow \big(a^{2} \times a^{2} + a^{2} \times b^{2} - a^{2} \times b^{2} \times - b^{2} \big) - \big(a^{2} + b^{2}\big)^{2}}$

$\sf{\Rightarrow \big(a^{4} + \cancel{a^{2}b^{2}} - \cancel{a^{2}b^{2}} - b^{4} \big) - \big(a^{2} + b^{2}\big)^{2}}$

$\sf{\Rightarrow \big(a^{4} - b^{4} \big) - \big(a^{2} + b^{2}\big)^{2}}$

$\sf{\Rightarrow \big(a^{4} - b^{4} \big) - \left(\big(a^{2}\big)^{2} - 2 \times a^{2} \times b^{2} + \big(b^{2}\big)^{2}\right)}$

$\sf{\Rightarrow \big(a^{4} - b^{4} \big) - \left(a^{4} - 2a^{2}b^{2} + b^{4}\right)}$

$\sf{\Rightarrow \cancel{a^{4}} - b^{4} - \cancel{a^{4}} + 2a^{2}b^{2} - b^{4}}$

$\sf{\Rightarrow - b^{4} + 2a^{2}b^{2} - b^{4}}$

$\purple{\sf{\Rightarrow 2a^{2}b^{2} - 2b^{4}}}$

Hence , $\green{\sf{\underline{\boxed{\therefore \big(a^{2} - b^{2}\big)\big(a^{2} + b^{2}\big) - \big(a^{2} + b^{2}\big)^{2} = 2a^{2}b^{2} - 2b^{4}}}}}$

Alternative Method :

$\purple{\sf{\big(a^{2} - b^{2}\big)\big(a^{2} + b^{2}\big) - \big(a^{2} + b^{2}\big)^{2}}}$

Using the identity ,

$\sf{a^{2} - b^{2} = (a + b)(a - b)}$

$\sf{a^{2} - b^{2} = a^{2} + b^{2} - 2ab}$

We Get :

$\sf{\Rightarrow \big(a^{2}\big)^{2} - \big(b^{2}\big)^{2} - \big(a^{2} + b^{2}\big)^{2}}$

$\sf{\Rightarrow a^{4} - b^{4} - \big(a^{2} + b^{2}\big)^{2}}$

$\sf{\Rightarrow a^{4} - b^{4} - \left(\big(a^{2} \big)^{2} - 2 \times a^{2} \times b^{2} + \big(b^{2}\big)^{2}\right)}$

$\sf{\Rightarrow \cancel{a^{4}} - b^{4} - \cancel{a^{4}} + 2a^{2}b^{2} - b^{4}}$

$\sf{\Rightarrow - b^{4} + 2a^{2}b^{2} - b^{4}}$

$\purple{\sf{\Rightarrow 2a^{2}b^{2} - 2b^{4}}}$

Hence ,

$\green{\sf{\underline{\boxed{\therefore \big(a^{2} - b^{2}\big)\big(a^{2} + b^{2}\big) - \big(a^{2} + b^{2}\big)^{2} = 2a^{2}b^{2} - 2b^{4}}}}}$

» Additional information :

• $\sf{(a + b)^{2} = a^{2} + b^{2} + 2ab}$

• $\sf{a^{2} + b^{2} = (a + b)^{2} - 2ab}$

• $\sf{a^{2} + b^{2} = (a - b)^{2} + 2ab}$